[FOM] Improving the Fundamental Theorem of Algebra
joeshipman at aol.com
Sun Nov 27 01:58:48 EST 2005
Ever since Gauss, it has been known that if every odd degree polynomial
in a field K has a root, and every element of K has a square root in
K(i), then K(i) is algebraically closed.
I recently proved that "odd degree" may be weakened to "odd prime
degree" here, and that this is the best possible (for any prime p,
there are fields in which all polynomials with degree not divisible by
p have roots, but some of degree p don't).
This proof has been reviewed by other mathematicians and I am writing
it up for publication.
I'm not claiming that this is a very deep result; the proof is only 2
pages long and quite elementary, depending only on very basic facts
about symmetric polynomials, binomial coefficients, and algebraic
What I'm interested in discussing on the FOM list is this:
If this theorem was ever proved before, why do ALL textbooks on algebra
require all odd degrees in their treatment of real closed fields, and
not mention, even as an exercise or in a footnote, that this assumption
is stronger than necessary?
If this theorem was never proved before, why not? It's an obvious
question to ask whether the hypothesis "Odd degree polynomials have
roots" can be weakened, especially since the theorem for odd degrees
has been regarded as the "algebraic part" of the Fundamental Theorem of
Algebra, which generalizes to all fields of characteristic 0. (The
"analytic" or "topological" part has been, at a minimum, that real
polynomials of odd degree have roots in the real numbers; if you don't
push the algebra as far, you need more analysis/topology, such as a
winding number argument or an application of Liouville's theorem).
I'll be happy to email the proof to anyone who wishes to see it.
-- Joseph Shipman
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